N-gram Introduction to N-gram

Language Language Models

Modeling
Predicting words
The water of Walden Pond is beautifully ...

blue
*refrigerator
green
*that
clear
Language Models
Systems that can predict upcoming words
• Can assign a probability to each potential next word
• Can assign a probability to a whole sentence
Why word prediction?
It's a helpful part of language tasks
• Grammar or spell checking
Their are two midterms Their There are two midterms
Everything has improve Everything has improve
improved

• Speech recognition
I will be back soonish I will be bassoon dish
Why word prediction?
It's how large language models (LLMs) work!
LLMs are trained to predict words
• Left-to-right (autoregressive) LMs learn to predict next word
LLMs generate text by predicting words
• By predicting the next word over and over again
Language Modeling (LM) more
formally
Goal: compute the probability of a sentence or
sequence of words W:
P(W) = P(w1,w2,w3,w4,w5…wn)
Related task: probability of an upcoming word:
P(w5|w1,w2,w3,w4) or P(wn|w1,w2…wn-1)
An LM computes either of these:
P(W) or P(wn|w1,w2…wn-1)
How to estimate these
probabilities
Could we just count and divide?

No! Too many possible sentences!

We’ll never see enough data for estimating these
How to compute P(W) or P(wn|w1, …wn-1)

How to compute the joint probability P(W):

P(The, water, of, Walden, Pond, is, so, beautifully, blue)

Intuition: let’s rely on the Chain Rule of Probability

Reminder: The Chain Rule
Recall the definition of conditional probabilities
P(B|A) = P(A,B)/P(A)Rewriting: P(A,B) = P(A) P(B|A)

More variables:
P(A,B,C,D) = P(A) P(B|A) P(C|A,B) P(D|A,B,C)
The Chain Rule in General
P(x1,x2,x3,…,xn) = P(x1)P(x2|x1)P(x3|x1,x2)…P(xn|x1,…,xn-1)
The Chain Rule applied to compute joint
probability of words in sentence

P(“The water of Walden Pond”) =

P(The) × P(water|The) × P(of|The water)
× P(Walden|The water of) × P(Pond|The water of Walden
Markov Assumption

Simplifying assumption:
Andrei Markov

Wikimedia commons
Bigram Markov Assumption

Instead of:
More generally, we approximate each
component in the product
 Simplest case: Unigram model

Some automatically generated sentences from two different unigram models

To him swallowed confess hear both . Which . Of save on trail
for are ay device and rote life have

Hill he late speaks ; or ! a more to leg less first you enter

Months the my and issue of year foreign new exchange’s September

were recession exchange new endorsed a acquire to six executives

 Bigram model

Some automatically generated sentences rom two different unigram models

Why dost stand forth thy canopy, forsooth; he is this palpable hit
the King Henry. Live king. Follow.

What means, sir. I confess she? then all sorts, he is trim, captain.

Last December through the way to preserve the Hudson corporation N.

B. E. C. Taylor would seem to complete the major central planners one
gram point five percent of U. S. E. has already old M. X. corporation
of living

on information such as more frequently fishing to keep her

Problems with N-gram models
• N-grams can't handle long-distance dependencies:
“The soups that I made from that new cookbook I
bought yesterday were amazingly delicious."
• N-grams don't do well at modeling new sequences
with similar meanings
The solution: Large language models
•can handle much longer contexts
•because of using embedding spaces, can model
synonymy better, and generate better novel strings
Why N-gram models?
A nice clear paradigm that lets us introduce many of
the important issues for large language models
•training and test sets
•the perplexity metric
•sampling to generate sentences
•ideas like interpolation and backoff
N-gram Introduction to N-grams
Language
Modeling
N-gram Estimating N-gram
Language Probabilities
Modeling
Estimating bigram probabilities
The Maximum Likelihood Estimate
An example
<s> I am Sam </s>
<s> Sam I am </s>
<s> I do not like green eggs and ham </s>
More examples:
Berkeley Restaurant Project sentences
can you tell me about any good cantonese restaurants close by
tell me about chez panisse
i’m looking for a good place to eat breakfast
when is caffe venezia open during the day
Raw bigram counts
Out of 9222 sentences
Raw bigram probabilities
Normalize by unigrams:

Result:
Bigram estimates of sentence probabilities

P(<s> I want english food </s>) =

P(I|<s>)
× P(want|I)
× P(english|want)
× P(food|english)
× P(</s>|food)
= .000031
What kinds of knowledge do N-grams represent?

P(english|want) = .0011
P(chinese|want) = .0065
P(to|want) = .66
P(eat | to) = .28
P(food | to) = 0
P(want | spend) = 0
P (i | <s>) = .25
Dealing with scale in large n-
grams

LM probabilities are stored and computed in

log format, i.e. log probabilities
This avoids underflow from multiplying many
small numbers

If we need probabilities we can do one exp at the end

Larger ngrams
4-grams, 5-grams
Large datasets of large n-grams have been released
• N-grams from Corpus of Contemporary American
English (COCA) 1 billion words (Davies 2020)
• Google Web 5-grams (Franz and Brants 2006) 1 trillion words)
• Efficiency: quantize probabilities to 4-8 bits instead of 8-byte float
Newest model: infini-grams (∞-grams) (Liu et al 2024)
• No precomputing! Instead, store 5 trillion words of web text in
suffix arrays. Can compute n-gram probabilities with any n!
N-gram LM Toolkits

SRILM
◦ http://www.speech.sri.com/projects/srilm/
KenLM
◦ https://kheafield.com/code/kenlm/
N-gram Estimating N-gram
Language Probabilities
Modeling
Languag Evaluation and Perplexity
e
Modelin
g
How to evaluate N-gram models

"Extrinsic (in-vivo) Evaluation"

To compare models A and B
1. Put each model in a real task
• Machine Translation, speech recognition, etc.
2. Run the task, get a score for A and for B
• How many words translated correctly
• How many words transcribed correctly
3. Compare accuracy for A and B
Intrinsic (in-vitro) evaluation
Extrinsic evaluation not always possible
• Expensive, time-consuming
• Doesn't always generalize to other applications
Intrinsic evaluation: perplexity
• Directly measures language model performance at
predicting words.
• Doesn't necessarily correspond with real application
performance
• But gives us a single general metric for language models
• Useful for large language models (LLMs) as well as n-grams
Training sets and test sets
We train parameters of our model on a training set.
We test the model’s performance on data we
haven’t seen.
◦ A test set is an unseen dataset; different from training set.
◦ Intuition: we want to measure generalization to unseen data
◦ An evaluation metric (like perplexity) tells us how well
our model does on the test set.
Choosing training and test sets

• If we're building an LM for a specific task

• The test set should reflect the task language we
want to use the model for
• If we're building a general-purpose model
• We'll need lots of different kinds of training data
• We don't want the training set or the test set to
be just from one domain or author or language.
Training on the test set
We can’t allow test sentences into the training set
• Or else the LM will assign that sentence an artificially
high probability when we see it in the test set
• And hence assign the whole test set a falsely high
probability.
• Making the LM look better than it really is
This is called “Training on the test set”
Bad science!
35
Dev sets
• If we test on the test set many times we might
implicitly tune to its characteristics
• Noticing which changes make the model better.
• So we run on the test set only once, or a few times
• That means we need a third dataset:
• A development test set or, devset.
• We test our LM on the devset until the very end
• And then test our LM on the test set once
Intuition of perplexity as evaluation metric: How
good is our language model?

Intuition: A good LM prefers "real" sentences

• Assign higher probability to “real” or “frequently
observed” sentences
• Assigns lower probability to “word salad” or
“rarely observed” sentences?
 Intuition of perplexity 2:
Predicting upcoming words
time 0.9
The Shannon Game: How well can we predict
dream 0.03
the next word?
• Once upon a ____ midnight 0.02
• That is a picture of a ____ …
• For breakfast I ate my usual ____ and 1e-100

Claude Shannon
Unigrams are terrible at this game (Why?)

A good LM is one that assigns a higher probability

to the next word that actually occurs
Picture credit: Historiska bildsamlingen
https://creativecommons.org/licenses/by/2.0/
Intuition of perplexity 3: The best language model is
one that best predicts the entire unseen test set

• We said: a good LM is one that assigns a higher

probability to the next word that actually occurs.
• Let's generalize to all the words!
• The best LM assigns high probability to the entire test
set.
• When comparing two LMs, A and B
• We compute PA(test set) and PB(test set)
• The better LM will give a higher probability to (=be less
surprised by) the test set than the other LM.
Intuition of perplexity 4: Use perplexity
instead of raw probability
• Probability depends on size of test set
• Probability gets smaller the longer the text
• Better: a metric that is per-word, normalized by length
• Perplexity is the inverse probability of the test set,
normalized by the number of words
Intuition of perplexity 5: the inverse
Perplexity is the inverse probability of the test set, normalized
by the number of words

(The inverse comes from the original definition of perplexity

from cross-entropy rate in information theory)
Probability range is [0,1], perplexity range is [1,∞]
Minimizing perplexity is the same as maximizing probability
Intuition of perplexity 6: N-grams

Chain rule:

Bigrams:
 Intuition of perplexity 7:
Weighted average branching factor
Perplexity is also the weighted average branching factor of a language.
Branching factor: number of possible next words that can follow any word
Example: Deterministic language L = {red,blue, green}
Branching factor = 3 (any word can be followed by red, blue, green)
Now assume LM A where each word follows any other word with equal probability ⅓
Given a test set T = "red red red red blue"
PerplexityA(T) = PA(red red red red blue)-1/5 = ) )⅓((
1/5- 5
)⅓( = 3=
1-

But now suppose red was very likely in training set, such that for LM B:
◦ P(red) = .8 p(green) = .1 p(blue) = .1
We would expect the probability to be higher, and hence the perplexity to be smaller:
PerplexityB(T) = PB(red red red red blue)-1/5
= (.8 * .8 * .8 * .8 * .1) -1/5 =.04096 -1/5 = .527-1 = 1.89
Holding test set constant:
Lower perplexity = better language model

Training 38 million words, test 1.5 million words, WSJ

N-gram Unigram Bigram Trigram

Order
Perplexity 962 170 109
Languag Evaluation and Perplexity
e
Modelin
g
Languag Sampling and Generalization
e
Modelin
g
The Shannon (1948) Visualization Method
Sample words from an LM
Claude Shannon
Unigram:
REPRESENTING AND SPEEDILY IS AN GOOD APT OR COME
CAN DIFFERENT NATURAL HERE HE THE A IN CAME THE TO
OF TO EXPERT GRAY COME TO FURNISHES THE LINE
MESSAGE HAD BE THESE.

Bigram:
THE HEAD AND IN FRONTAL ATTACK ON AN ENGLISH WRITER
THAT THE CHARACTER OF THIS POINT IS THEREFORE
ANOTHER METHOD FOR THE LETTERS THAT THE TIME OF WHO
EVER TOLD THE PROBLEM FOR AN UNEXPECTED.
 How Shannon sampled those words in 1948

"Open a book at random and select a letter at random on the

page. This letter is recorded. The book is then opened to another
page and one reads until this letter is encountered. The
succeeding letter is then recorded. Turning to another page this
second letter is searched for and the succeeding letter recorded,
etc."
Sampling a word from a
distribution
 Visualizing Bigrams the Shannon Way

Choose a random bigram (<s>, w)

<s> I
according to its probability p(w|<s>) I want
Now choose a random bigram (w, x) want to
according to its probability p(x|w) to eat
And so on until we choose </s> eat Chinese
Chinese food
Then string the words together
food </s>
I want to eat Chinese food
Note: there are other sampling
methods
Used for neural language models
Many of them avoid generating words from the very
unlikely tail of the distribution
We'll discuss when we get to neural LM decoding:
◦ Temperature sampling
◦ Top-k sampling
◦ Top-p sampling
Approximating Shakespeare
Shakespeare as corpus

N=884,647 tokens, V=29,066

Shakespeare produced 300,000 bigram types out of
V2= 844 million possible bigrams.
◦ So 99.96% of the possible bigrams were never seen (have
zero entries in the table)
◦ That sparsity is even worse for 4-grams, explaining why
our sampling generated actual Shakespeare.
The Wall Street Journal is not
Shakespeare
Can you guess the author? These 3-gram
sentences are sampled from an LM trained on who?
1) They also point to ninety nine point six
billion dollars from two hundred four oh
six three percent of the rates of interest
stores as Mexico and gram Brazil on market
conditions
2) This shall forbid it should be branded,
if renown made it empty.
3) “You are uniformly charming!” cried he,
with a smile of associating and now and
then I bowed and they perceived a chaise
and four to wish for. 55
Choosing training data
If task-specific, use a training corpus that has a similar
genre to your task.
• If legal or medical, need lots of special-purpose documents
Make sure to cover different kinds of dialects and
speaker/authors.
• Example: African-American Vernacular English (AAVE)
• One of many varieties that can be used by African Americans and others
• Can include the auxiliary verb finna that marks immediate future tense:
• "My phone finna die"
The perils of overfitting
N-grams only work well for word prediction if the
test corpus looks like the training corpus
• But even when we try to pick a good training
corpus, the test set will surprise us!
• We need to train robust models that generalize!
One kind of generalization: Zeros
• Things that don’t ever occur in the training set
• But occur in the test set
 Zeros
Training set: • Test set
… ate lunch … ate lunch
… ate dinner … ate breakfast
… ate a
… ate the
P(“breakfast” | ate) = 0
Zero probability bigrams
Bigrams with zero probability
◦ Will hurt our performance for texts where those words
appear!
◦ And mean that we will assign 0 probability to the test set!
And hence we cannot compute perplexity (can’t
divide by 0)!
Languag Sampling and Generalization
e
Modelin
g
N-gram Smoothing, Interpolation,
Language and Backoff
Modeling
The intuition of smoothing (from Dan Klein)

When we have
P(w | denied the)
sparse statistics:

allegations
3 allegations

outcome
reports
2 reports

attack
…

request
claims
1 claims

man
1 request
7 total

Steal probability
P(w | denied the)
mass to generalize better
2.5 allegations

allegations
allegations
1.5 reports

outcome
0.5 claims

reports

attack
0.5 request
…

man
claims

request
2 other
7 total
Add-one estimation
Also called Laplace smoothing
Pretend we saw each word one more time than we did
Just add one to all the counts!

MLE estimate:

Add-1 estimate:
Maximum Likelihood Estimates
The maximum likelihood estimate
◦ of some parameter of a model M from a training set T
◦ maximizes the likelihood of the training set T given the model M

Suppose the word “bagel” occurs 400 times in a corpus of a million words
What is the probability that a random word from some other text will be
“bagel”?
MLE estimate is 400/1,000,000 = .0004
This may be a bad estimate for some other corpus
◦ But it is the estimate that makes it most likely that “bagel” will occur 400
times in a million word corpus.
Berkeley Restaurant Corpus: Laplace smoothed
bigram counts
Laplace-smoothed bigrams
Reconstituted counts
Compare with raw bigram
counts
Add-1 estimation is a blunt
instrument
So add-1 isn’t used for N-grams:
◦ Generally we use interpolation or backoff instead
But add-1 is used to smooth other NLP models
◦ For text classification
◦ In domains where the number of zeros isn’t so huge.
Backoff and Interpolation
Sometimes it helps to use less context
◦ Condition on less context for contexts you know less about
Backoff:
◦ use trigram if you have good evidence,
◦ otherwise bigram, otherwise unigram
Interpolation:
◦ mix unigram, bigram, trigram

Interpolation works better

Linear Interpolation
Simple interpolation

Lambdas conditional on context:

How to set λs for interpolation?
Use a held-out corpus
Held-Out Test
Training Data Data Data

Choose λs to maximize probability of held-out data:

◦ Fix the N-gram probabilities (on the training data)
◦ Then search for λs that give largest probability to held-
out set
Backoff
Suppose you want:
P(pancakes| delicious soufflé)
If the trigram probability is 0, use the bigram
P(pancakes| soufflé)
If the bigram probability is 0, use the unigram
P(pancakes)
Complication: need to discount the higher-order ngram so
probabilities don't sum higher than 1 (e.g., Katz backoff)
Stupid Backoff
Backoff without discounting (not a true probability)

74
N-gram Interpolation and Backoff
Language
Modeling